Spirality and Rectilinear Planarity Testing of Independent-Parallel SP-Graphs

TL;DR

This paper presents a linear-time algorithm for testing rectilinear planarity of independent-parallel SP-graphs, overcoming challenges posed by higher vertex degrees where previous methods failed.

Contribution

It introduces a novel approach using spirality properties to efficiently test rectilinear planarity in a broader class of SP-graphs.

Findings

Linear-time rectilinear planarity testing algorithm for independent-parallel SP-graphs

Logarithmic lower bound on spirality for degree-four vertices in SP-graphs

Extension of planarity testing techniques beyond degree-three vertex restrictions

Abstract

We study the long-standing open problem of efficiently testing rectilinear planarity of series-parallel graphs (SP-graphs) in the variable embedding setting. A key ingredient behind the design of a linear-time testing algorithm for SP-graphs of vertex-degree at most three is that one can restrict the attention to a constant number of ``rectilinear shapes'' for each series or parallel component. To formally describe these shapes the notion of spirality can be used. This key ingredient no longer holds for SP-graphs with vertices of degree four, as we prove a logarithmic lower bound on the spirality of their components. The bound holds even for the independent-parallel SP-graphs, in which no two parallel components share a pole. Nonetheless, by studying the spirality properties of the independent-parallel SP-graphs, we are able to design a linear-time rectilinear planarity testing algorithm for this graph family.